> On Thursday, March 14, 2013 2:03:35 PM UTC-7, Jesse F. Hughes wrote:>>> Is the negation of Fermat's theorem a falsifiable statement? If so,>> how might one show that it is false?>>> Didn't you just answer your own question?

No.

>> Is it the case that if a given mathematical statement is>> "falsifiable", then so is its negation? >>> I think you've already answered your own question.

We don't seem to have the same ideas of falsifiability[1].

Here's my intuitions. The statement

All ravens are black.

is falsifiable, since the discovery of a white raven would prove itfalse.

The statement

Not all ravens are black.

is not falsifiable, except in exceptional circumstances[2]. If I seemany ravens and all of them are black, then I still can't concludethat this statement is false, since the next raven I see might bewhite. So, although the statement is verifiable, it is notfalsifiable.

Obviously,

(An>2) NOT (E a,b,c)( a^n+b^n=c^n )

is like the first statement and hence plausibly falsifiable, while itsnegation is like the second statement and hence not falsifiable.

But maybe I'm wrong. How would you set out to prove that

Not all ravens are black.

is false?

Footnotes: [1] Here, I assume we're using the usual, fairly naive notions offalsifiability which actually have not been fashionable in philosophyof science for quite some time, due to a number of issues which neednot detain us.

[2] If there is a finite number of ravens and you know when you havefound them all, then it is falsifiable.

-- Jesse F. Hughes"It is not as satisfying to disagree with a book." -- Russell Easterly, on why he argues against set theory without reading a book on set theory.